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Theorem releq 4422
Description: Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
releq  |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )

Proof of Theorem releq
StepHypRef Expression
1 sseq1 2966 . 2  |-  ( A  =  B  ->  ( A  C_  ( _V  X.  _V )  <->  B  C_  ( _V 
X.  _V ) ) )
2 df-rel 4352 . 2  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
3 df-rel 4352 . 2  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
41, 2, 33bitr4g 212 1  |-  ( A  =  B  ->  ( Rel  A  <->  Rel  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    = wceq 1243   _Vcvv 2557    C_ wss 2917    X. cxp 4343   Rel wrel 4350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-rel 4352
This theorem is referenced by:  releqi  4423  releqd  4424  dfrel2  4771  tposfn2  5881  ereq1  6113
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